Answer
$x=-\dfrac{1}{5}$
Work Step by Step
Multiplying both sides by the $LCD,
6
,$ the value of the variable that satisfies the given expression, $
\dfrac{2}{3}\left( \dfrac{1}{2}-x \right)+\dfrac{5}{6}=\dfrac{3}{2}\left( \dfrac{2}{3}x+1 \right)
,$ is
\begin{array}{l}\require{cancel}
2\cdot2\left( \dfrac{1}{2}-x \right)+1(5)=3\cdot3\left( \dfrac{2}{3}x+1 \right)
\\\\
4\left( \dfrac{1}{2}-x \right)+5=9\left( \dfrac{2}{3}x+1 \right)
\\\\
\left( \dfrac{4}{2}-4x \right)+5=\dfrac{18}{3}x+9(1)
\\\\
2-4x+5=6x+9
\\\\
-4x-6x=9-2-5
\\\\
-10x=2
\\\\
x=\dfrac{2}{-10}
\\\\
x=-\dfrac{1}{5}
.\end{array}